
with \((\mu _0, \mu _1)\)  the unknown arm means parameters, we have that TS's allocation probabilities [1]}, [2]} at each step \(t\)  and for each arm are given by:
\(\tilde{\pi }_{t,1}^{\text{TS}} &= \mathbb {P}(\tilde{\mu }_{t,1} > \tilde{\mu }_{t,0} | \mathcal {H}_{t-1}) = \mathbb {P}(\tilde{\mu }_{t,1} - \tilde{\mu }_{t,0} > 0 | \mathcal {H}_{t-1}) = \Phi \left(\frac{\mu _{D_t}}{\sigma _{D_t}} \Bigg | \mathcal {H}_{t-1}\right)\\\tilde{\pi }_{t,0}^{\text{TS}} &= \mathbb {P}(\tilde{\mu }_{t,0} > \tilde{\mu }_{t,1} | \mathcal {H}_{t-1}) = 1 - \tilde{\pi }_{t,1}^{\text{TS}} = 1 - \Phi \left(\frac{\mu _{D_t}}{\sigma _{D_t}} \Bigg | \mathcal {H}_{t-1}\right) = \Phi \left(-\frac{\mu _{D_t}}{\sigma _{D_t}} \Bigg | \mathcal {H}_{t-1}\right),\) 
